Consider the following minimization problem: $$\min_{x\in\mathbb{R}^n}f(x),$$ where $f$ is a proper convex closed function and suppose that there exists at least one solution of the above-mentioned problem. Proximal point algorithm works choosing an initial point $x_0$ and the minimizing sequence is defined recursively by $$x_k=\arg\,\min_{x\in\mathbb{R}^n}f(x)+\frac{1}{2a_k}\lVert x−x_{k−1}\rVert^2,\quad\text{for any }k∈N.$$
By proximal algorithm theory, the sequence $(x_k)_k$ converges to a minimum point of $f$. There exists a connection between the limit point of $(x_k)$ and the initial point $x_0$? For example, in the Tikhonov regularization the minimizing sequence converges to the unique minimum norm element in the the set of the minimum points of $f$. There exists an analogous characterzation even for proximal algorithms?